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Trigonometry Sec. 05 notes MathHands.com Márquez Famous IDs: Pythagoras Identities Main Idea The previous section was extremely important for a few reasons. First, you should now be comfortable understanding what an identity is. Second, we saw methods used to establish identities, working on each side independently, tweaking a known identity, graphing the functions and comparing the graphs, and last, proving identities by starting with a very creative idea. Moreover, we introduced the first dose of fundamental and famous trigonometric identities, most or all of which were proved in the corresponding assignment. Now, the task at hand is to expand the list of very famous trigonometric identities, and to practice our proving skills. Below is our next dose of famous identities, the first of which is the most famous of them all, the pythagoras identities. Pythagoras Identities sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ sin2 θ = 1 − cos2 θ cos2 θ = 1 − sin2 θ tan2 θ = sec2 θ − 1 cot2 θ + 1 = csc2 θ p sin θ = ± 1 − cos2 θ p sec θ = ± tan2 θ + 1 cot2 θ = csc2 θ − 1 p tan θ = ± sec2 θ − 1 EXAMPLE 1 (work on one side) The first identity in this family is the most famous trigonometric identity. We will prove it by working on one side. Without loss of generality, we can assume the following is a reference triangle for the angle θ c b θ a sin2 θ + cos2 θ b 2 c b2 c2 + + b2 +a2 c2 c2 c2 1 c 2007-2009 MathHands.com a2 c2 a 2 c ? = | | | | | | | | | = 1 want to know definitions of sine and cosine algebra clean up algebra clean up Pythagoras Theorem, see reference triangle 1 algebra clean up math hands pg. 1 Trigonometry Sec. 05 notes MathHands.com Márquez EXAMPLE 2 (tweak a known identity) Prove the following is an identity. tan2 θ + 1 = sec2 θ Solution: We will prove this identity by tweaking the previous one, the Pythagoras Identity. sin2 θ + cos2 sin2 θ + cos2 1 cos2 θ sin2 θ cos2 θ = 1 + sin θ 2 cos θ cos2 θ cos2 θ + 2 cos θ 2 cos θ 2 We already know this one, proven above! = 1 cos2 θ = 1 cos2 θ = (1) algebra 2 1 cos θ (tan θ) + (1) = (sec θ) tan2 θ + 1 = sec2 θ c 2007-2009 MathHands.com multiply both sides 2 algebra proven identity done! math hands pg. 2 Trigonometry Sec. 05 exercises MathHands.com Márquez Famous IDs: Pythagoras Identities 1. Prove and OWN [VERY famous identity]. sin2 θ + cos2 θ = 1 2. Prove and OWN [VERY famous identity]. tan2 θ + 1 = sec2 θ 3. Prove and OWN [VERY famous identity]. tan2 θ = sec2 θ − 1 4. Prove and OWN [VERY famous identity]. cot2 θ + 1 = csc2 θ 5. Prove and OWN [VERY famous identity]. cot2 θ = csc2 θ − 1 6. Prove the following famous identity sin2 θ = 1 − cos2 θ 7. Prove and OWN [VERY famous identity]. p sin θ = ± 1 − cos2 θ 8. Prove and OWN [VERY famous identity]. tan θ = ± p sec2 θ − 1 9. Prove and OWN [VERY famous identity]. p sec θ = ± tan2 θ + 1 10. Prove the following famous identity tan2 θ = (sec θ − 1)(sec θ + 1) 11. Prove the following is an identity OR prove the following is not an identity 1 = sec2 θ + tan θ sec θ 1 − sin θ c 2007-2009 MathHands.com math hands pg. 3 Trigonometry Sec. 05 exercises MathHands.com Márquez 12. Prove the following is an identity OR prove the following is not an identity 1 = cot2 θ + cot2 θ sec θ 1 − sec θ 13. Prove the following is an identity OR prove the following is not an identity 1 1 + = 2 sec2 θ 1 − sin θ 1 + sin θ 14. Prove the following is an identity OR prove the following is not an identity cos4 θ = 1 − 2 sin2 + sin4 θ 15. Prove the following is an identity OR prove the following is not an identity cos6 θ = 1 − 3 sin2 θ + 3 sin4 θ − sin6 θ 16. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. 2x2 − 1 = 1 − 2y 2 A. TRUE B. FALSE 17. Practice Work on each side: Determine if the following is an identity, prove your answer 2 cos2 x − 1 = 1 − 2 sin2 x A. TRUE B. FALSE 18. Practice Work on each side: Assume there is an interesting world in which x2 + y 2 = 1 for all values of x and y. In such world, Determine if the following is an identity, if so answer TRUE, if not answer FALSE. (x + y)2 = 2xy + 1 A. TRUE B. FALSE Therefore TRUE 19. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)2 = 2 cos x sin x + 1 A. TRUE B. FALSE Therefore TRUE c 2007-2009 MathHands.com math hands pg. 4 Trigonometry Sec. 05 exercises MathHands.com Márquez 20. Practice Work on each side: Determine if the following is an identity, prove your answer. cos4 x − sin4 x = 2 cos4 x − 2 cos2 x + 1 A. TRUE B. FALSE 21. Practice Work on each side: Determine if the following is an identity, prove your answer. (cos x + sin x)4 = −4 cos4 x + 4 cos2 x + 4 cos x sin x + 1 A. TRUE B. FALSE 22. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 cos x + sin x cos x − sin x = A. TRUE B. FALSE 23. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 1 − 2 sin2 x A. TRUE B. FALSE 24. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 2 cos2 x − 1 A. TRUE B. FALSE 25. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 1 = 2 2x−1 2 cos 1 − 2 sin x A. TRUE B. FALSE 26. Practice Work on each side: Determine if the following is an identity, prove your answer. cos x − sin x = A. TRUE 1 2 cos x + sin x B. FALSE 27. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 cos x 1 − = cos x + sin x cos x − sin x 1 − 2 sin2 x A. TRUE B. FALSE 28. Practice Work on each side: Determine if the following is an identity, prove your answer. 1 2 cos x 1 + = cos x + sin x cos x − sin x 1 − 2 cos2 x A. TRUE B. FALSE c 2007-2009 MathHands.com math hands pg. 5 Trigonometry Sec. 05 exercises MathHands.com Márquez 29. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x + 2 = 3 A. TRUE B. FALSE 30. Practice Work on each side: Determine if the following is an identity, prove your answer. cos6 x + 3 cos4 x sin2 x + 3 cos2 x sin4 x + sin6 x − cos2 x = sin2 x A. TRUE B. FALSE 31. Prove the following is an identity OR prove the following is not an identity cos6 θ = −2 + 3 cos2 θ + 3 sin4 θ − sin6 θ 32. Prove the following is an identity OR prove the following is not an identity 1 − cos4 θ = sin2 θ + sin2 θ cos2 θ 33. (**)Prove the identity sin(2x) = 2 sin x cos x c 2007-2009 MathHands.com math hands pg. 6